If ` x^(y) * y^(x) = (x + y) ^(5) , " find " (dy)/( dx)`

Answer

To solve the equation \( x^y \cdot y^x = (x + y)^5 \) and find \( \frac{dy}{dx} \), we will follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the natural logarithm of both sides of the equation:

\[ \ln(x^y \cdot y^x) = \ln((x + y)^5) \]

### Step 2: Apply logarithmic properties
Using the properties of logarithms, we can simplify both sides:

\[ \ln(x^y) + \ln(y^x) = 5 \ln(x + y) \]

This simplifies to:

\[ y \ln x + x \ln y = 5 \ln(x + y) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \). We will use the product rule on the left side:

\[ \frac{d}{dx}(y \ln x) + \frac{d}{dx}(x \ln y) = \frac{d}{dx}(5 \ln(x + y)) \]

Differentiating the left side:

1. For \( y \ln x \):
- \( \frac{dy}{dx} \ln x + y \cdot \frac{1}{x} \)

2. For \( x \ln y \):
- \( \ln y + x \cdot \frac{1}{y} \cdot \frac{dy}{dx} \)

So the left side becomes:

\[ \frac{dy}{dx} \ln x + \frac{y}{x} + \ln y + x \cdot \frac{1}{y} \cdot \frac{dy}{dx} \]

Differentiating the right side:

\[ \frac{5}{x + y} \left(1 + \frac{dy}{dx}\right) \]

### Step 4: Set the derivatives equal
Now we set the derivatives equal to each other:

\[ \frac{dy}{dx} \ln x + \frac{y}{x} + \ln y + \frac{x}{y} \frac{dy}{dx} = \frac{5}{x + y} \left(1 + \frac{dy}{dx}\right) \]

### Step 5: Collect terms involving \( \frac{dy}{dx} \)
Rearranging gives us:

\[ \frac{dy}{dx} \left(\ln x + \frac{x}{y} - \frac{5}{x + y}\right) = \frac{5}{x + y} - \frac{y}{x} - \ln y \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{\frac{5}{x + y} - \frac{y}{x} - \ln y}{\ln x + \frac{x}{y} - \frac{5}{x + y}} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{5 - \frac{y(x + y)}{x} - (x + y) \ln y}{(x + y)(\ln x + \frac{x}{y} - \frac{5}{x + y})} \]